CHICAGO — Rod R. Blagojevich, the ousted former governor of Illinois, was convicted on Tuesday of making false statements to the Federal Bureau of Investigation, but the jury in the corruption case against him also reported that it was hopelessly deadlocked on the 23 other counts against him.

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Judge James B. Zagel of Federal District Court accepted the jury’s verdict on the false-statements count on Tuesday and declared a mistrial on the remaining counts, the bulk of the case. Federal prosecutors said immediately that they would try Mr. Blagojevich once more.